POSITIVE SOLUTIONS OF SOME NONLOCAL IMPULSIVE BOUNDARY VALUE PROBLEM
arXiv:1304.0206v1 [math.CA] 31 Mar 2013
GENNARO INFANTE AND PAOLAMARIA PIETRAMALA Abstract. We prove new results on the existence of positive solutions for some impulsive differential equation subject to nonlocal boundary conditions. Our boundary conditions involve an affine functional given by a Stieltjes integral. These cover the well known multi-point boundary conditions that are studied by various authors.
1. Introduction Differential equations with impulses arise quite often in the study of different problems, in particular are used as a model for evolutionary processes subject to a sudden rapid change of their state at certain moments. The theory of impulsive differential equations has become recently a quite active area of research. For an introduction to this theory we refer to the books [4, 5, 19, 32], which also contain a variety of interesting examples and applications. More recently, boundary value problems (BVPs) for impulsive second-order differential equations have been studied by several authors, see for example [1, 2, 7, 8, 12, 13, 23, 24, 25, 26, 27, 31, 36] and the references therein. In particular, the existence of positive solutions under the so-called m-point boundary conditions has been investigated, in the context of impulsive differential equations, in [9, 10, 18]. Various techniques are utilized in the above papers: the Leggett-Williams theorem, the Schauder fixed point theorem, the method of upper and lower solutions, the fixed point index on cones and, of course, the well-known Guo-Krasnosel’ski˘ı theorem on cone-compression and cone-expansion. In this paper, we establish new results for the existence of positive solutions for the second order impulsive differential equation (1.1)
(1.2)
u′′ (t) + g(t)f (t, u(t)) = 0, t ∈ (0, 1), t 6= τ,
∆u|t=τ = I(u(τ )), ∆u′ |t=τ =
I(u(τ )) , τ −1
subject to the nonlocal boundary conditions (BCs) (1.3)
u(0) = α[u], u(1) = 0.
2000 Mathematics Subject Classification. Primary 34B37, secondary 34A37, 34B10, 34B18. Key words and phrases. Fixed point index, cone, positive solution, impulsive differential equation. 1
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GENNARO INFANTE AND PAOLAMARIA PIETRAMALA
Here τ ∈ (0, 1), ∆v|t=τ denotes the “jump” of v(t) in t = τ , that is ∆v|t=τ = v(τ + ) − v(τ − ), where v(τ − ), v(τ + ) are the left and right limits of v(t) in t = τ , and α[u] is a positive functional given by α[u] = A0 +
Z
1
u(s) dA(s),
0
involving a Lebesgue-Stieltjes integral. The impulsive differential equation (1.1)-(1.2), under different BCs has been studied in [13]. The type of BC we study here is quite general and includes as special cases α[u] =
m X
αi u(ξi ) and α[u] =
Z
1
α(s)u(s) ds,
0
i=1
that is, multi-point and integral BCs, that are widely studied objects. In the case of ordinary differential equations, this has been done in several papers, see for example [14, 17, 22, 28, 33, 34, 35] and the references therein. The methodology here is to write the boundary value problem (1.1)-(1.3) as a perturbed integral equation and we look for fixed points of an operator T in a suitable cone of positive functions in the space P C[0, 1]. One advantage of this approach is that we avoid lengthly calculations to determine the Green’s function associated to the impulsive BVP. For simplicity, we restrict our attention to the case of one impulse. In Remark 2.6 we suggest how this approach can be modified to work with a finite number of impulses. Our main ingredient is the classical fixed point index theory and in the last Section we provide an example to illustrate our theory. 2. Existence of positive solutions of some integral equations We study the existence of positive solutions of the integral equation Z 1 I(u(τ )) (2.1) u(t) = γ(t)α[u] + k(t, s)g(s)f (s, u(s)) ds + γ(t)χ(τ,1] , 1−τ 0 where t ∈ [0, 1], and τ ∈ (0, 1) is fixed. We work in the Banach space P C[0, 1] := {u : [0, 1] → R : u is continuous in t ∈ [0, 1]\{τ }, there exist u(τ − ) = u(τ ) and u(τ + ) < ∞}, endowed with the supremum norm kuk = sup{|u(t)| : t ∈ [0, 1]}. We make use of the classical fixed point index for compact maps (see for example [3] or [11]) on the cone (2.2) K = u ∈ P C[0, 1] : u(t) ≥ 0 ∀ t ∈ [0, 1] and min u(t) ≥ ckuk , t∈[a,b]
where [a, b] is some subset of (τ, 1) and c is a positive constant. From now on, we assume that f, g, α, γ, I and the kernel k have the following properties:
NONLOCAL IMPULSIVE BVPS
3
(C1 ) f : [0, 1] × [0, ∞) → [0, ∞) satisfies Carath´eodory conditions, that is, for each u, t 7→ f (t, u) is measurable and for almost every t, u 7→ f (t, u) is continuous, and for every r > 0 there exists a L∞ -function φr : [0, 1] → [0, ∞) such that f (t, u) ≤ φr (t) for almost all t ∈ [0, 1] and all u ∈ [0, r]. (C2 ) k : [0, 1] × [0, 1] → [0, ∞) is measurable, and for every t1 ∈ [0, 1] we have Z 1 lim |k(t, s) − k(t1 , s)|φr (s) ds = 0. t→t1
0
(C3 ) There exist [a, b] ⊂ (τ, 1), a L∞ -function Φ : [0, 1] → [0, ∞) and a constant c1 ∈ (0, 1] such that k(t, s) ≤ Φ(s) for t ∈ [0, 1] and almost every s ∈ [0, 1] k(t, s) ≥ c1 Φ(s) for t ∈ [a, b] and almost every s ∈ [0, 1]. (C4 ) γ : [0, 1] → [0, ∞) is continuous and there exists a constant c2 ∈ (0, 1] such that γ(t) ≥ c2 kγk for t ∈ [a, b]. (C5 ) α : K → [0, ∞) is a continuous functional with Z 1 α[u] = A0 + u(s) dA(s), 0
where dA is a positive Lebesgue-Stieltjes measure, A is continuous in τ and R1 dA(s) < ∞. 0 Rb (C6 ) g ∈ L1 [0, 1], g ≥ 0 a. e. and a Φ(s)g(s) ds > 0. (C7 ) I : [0, ∞) → [0, ∞) is a continuous function and there exist δ1 , δ2 ≥ 0 such that δ1 x ≤ I(x) ≤ δ2 x for x ∈ [0, ∞). Under these hypotheses we can work in the cone (2.2), where c = min{c1 , c2 } and [a, b] as in (C3 ). If Ω is a bounded open subset of K (in the relative topology) we denote by Ω and ∂Ω the closure and the boundary relative to K. We write Kr = {u ∈ K : kuk < r} and K r = {u ∈ K : kuk ≤ r}. We consider now the map T : K → P C[0, 1] defined, for u ∈ K, by Z 1 I(u(τ )) . T u(t) := γ(t)α[u] + k(t, s)g(s)f (s, u(s)) ds + γ(t)χ(τ,1] 1−τ 0 In order to prove that T is compact, we make use of the following compactness criterion, which can be found in [1, 19] and is an extension of the classical Ascoli-Arzel`a Theorem. A key ingredient here is that the interval is compact. For a compactness criterion on unbounded intervals and its applications to impulsive differential equations see [6, 29, 30].
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GENNARO INFANTE AND PAOLAMARIA PIETRAMALA
We recall that a set S ⊂ P C[0, 1] is said to be quasi-equicontinuous if for every u ∈ S and for every ε > 0 there exists δ > 0 such that t1 , t2 ∈ [0, τ ] (or t1 , t2 ∈ (τ, 1]) and |t1 − t2 | < δ implies |u(t1 ) − u(t2 )| < ε. Lemma 2.1. A set S ⊆ P C[0, 1] is relatively compact in P C[0, 1] if and only if S is bounded and quasi-equicontinuous. Theorem 2.2. If the hypotheses (C1 )-(C7 ) hold for some r > 0, then T maps K r into K. When these hypotheses hold for each r > 0, T maps K into K. Moreover, T is a compact map. Proof. Let u ∈ K r . Then we have, for t ∈ [0, 1], Z 1 I(u(τ )) T u(t) = γ(t) α[u] + χ(τ,1] + k(t, s)g(s)f (s, u(s)) ds ≥ 0, 1−τ 0 furthermore I(u(τ )) + T u(t) ≤ γ(t) α[u] + 1−τ
Z
1
Φ(s)g(s)f (s, u(s)) ds
0
and therefore we obtain (2.3)
Z 1 I(u(τ )) + Φ(s)g(s)f (s, u(s)) ds. kT uk ≤ kγk α[u] + 1−τ 0
Then we have Z 1 I(u(τ )) min T u(t) ≥ c2 kγk α[u] + + c1 Φ(s)g(s)f (s, u(s)) ds t∈[a,b] 1−τ 0 Z 1 h i I(u(τ )) ≥ c kγk α[u] + + Φ(s)g(s)f (s, u(s)) ds ≥ ckT uk. 1−τ 0 Hence T u ∈ K for every u ∈ K r . Now, we show that the map T is compact. Firstly, we show that T sends bounded sets into bounded sets. It is enough to see that T (K r ) is bounded. Let u ∈ K r . Then, for all t ∈ [0, 1], from (2.3) we have Z 1 δ2 r kT uk ≤ kγk α[u] + + Φ(s)g(s)φr (s) ds 1−τ 0 Z 1 δ2 r + Mr , ≤ kγk A0 + r dA(s) + 1−τ 0 for some 0 ≤ Mr < ∞. We prove now that T sends bounded sets into quasi-equicontinuous sets. It is sufficient to prove this for t1 , t2 ∈ (τ, 1], t1 < t2 and u ∈ K r . We have I(u(τ )) |T u(t1 ) − T u(t2 )| ≤|γ(t1 ) − γ(t2 )| α[u] + 1−τ Z 1 + |k(t1 , s) − k(t2 , s)|g(s)φr (s) ds. 0
Then |T u(t1) − T u(t2 )| → 0 when t1 → t2 . From Lemma 2.1 we can conclude that T is a compact map.
NONLOCAL IMPULSIVE BVPS
5
Let dB1 be the Dirac measure of weight δ1 /(1 − τ ) in τ and let dB2 be the Dirac measure of weight δ2 /(1 − τ ) in τ . We make use of the two functionals Z 1 Z 1 Z 1 α1 [u] := A0 + u(s) dA(s) + u(s)dB1 (s) := A0 + u(s) dA1 (s), 0
α2 [u] := A0 +
Z
0
1
u(s) dA(s) +
0
Z
0
1
u(s)dB2(s) := A0 + 0
Z
1
u(s) dA2(s) 0
and of the following numbers f 0,ρ :=
(2.4)
f (t, u) , ρ 0≤u≤ρ, 0≤t≤1
1 := sup m t∈[0,1]
sup
Z
fρ,ρ/c :=
1
f (t, u) , ρ≤u≤ρ/c, a≤t≤b ρ inf
1 := inf t∈[a,b] M(a, b)
k(t, s)g(s) ds,
0
Z
b
k(t, s)g(s) ds.
a
We assume that (C8 ) The function t 7→ k(t, s) is integrable with respect to the measure dA2 , that is Z 1 K(s) := k(t, s) dA2 (t) 0
is well defined. Firstly, we prove that the index is 1 on the set Kρ . Lemma 2.3. Suppose Γ :=
R1 0
γ(t) dA2 (t) < 1 and assume that there exists ρ > 0 such
that u 6= T u for all u ∈ ∂Kρ and (I1ρ ) the following inequality holds: (2.5)
kγk A0 kγk + f 0,ρ (1 − Γ)ρ (1 − Γ)
Z
1
K(s)g(s) ds + 0
1 ≤ 1. m
Then the fixed point index, iK (T, Kρ ), is 1. Proof. We show that λu 6= T u for every u ∈ ∂Kρ and for every λ > 1. In fact, if there exists λ > 1 and u ∈ ∂Kρ such that λu = T u then Z 1 I(u(τ )) λu(t) = γ(t) α[u] + χ(τ,1] + k(t, s)g(s)f (s, u(s)) ds 1−τ 0 Z 1 δ2 (u(τ )) + k(t, s)g(s)f (s, u(s)) ds. ≤ γ(t) α[u] + 1−τ 0 Then we have (2.6)
λu(t) ≤ γ(t)α2 [u] +
and λ
Z
0
Z
1
k(t, s)g(s)f (s, u(s)) ds 0
1
u(t)dA2 (t) ≤ α2 [u]Γ +
Z
0
1
K(s)g(s)f (s, u(s)) ds.
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GENNARO INFANTE AND PAOLAMARIA PIETRAMALA
Hence we obtain (λ − Γ)α2 [u] ≤ λA0 +
Z
1
K(s)g(s)f (s, u(s)) ds.
0
Substituting into (2.6) gives λA0 γ(t) γ(t) λu(t) ≤ + λ−Γ λ−Γ
Z
1
K(s)g(s)f (s, u(s)) ds + 0
Z
1
k(t, s)g(s)f (s, u(s)) ds.
0
Taking the supremum for t ∈ [0, 1] gives Z 1 Z 1 kγk λA0 kγk 0,ρ K(s)g(s)ρf ds + sup + k(t, s)g(s)ρf 0,ρ ds λρ ≤ λ−Γ λ−Γ 0 t∈[0,1] 0 Z 1 Z 1 A0 kγk kγk 0,ρ < + K(s)g(s)ρf ds + sup k(t, s)g(s)ρf 0,ρ ds. 1−Γ 1−Γ 0 t∈[0,1] 0 Thus we have, kγk A0 kγk + f 0,ρ λ< (1 − Γ)ρ (1 − Γ)
Z
1
K(s)g(s) ds + 0
1 ≤ 1. m
This contradicts the fact that λ > 1 and proves the result. We make use of the open set Vρ = u ∈ K : min u(t) < ρ . t∈[a,b]
Vρ is similar to the set called Ωρ/c in [21]. Note that Kρ ⊂ Vρ ⊂ Kρ/c . We now prove that the index is 0 on the set Vρ . Lemma 2.4. Assume that there exists ρ > 0 such that u 6= T u for u ∈ ∂Vρ and (I0ρ ) the following inequalities hold: (2.7)
α1 [u] ≥ α0 ρ for u ∈ ∂Vρ , where α0 ≥ 0, and
(2.8)
c2 kγkα0 +
1 fρ,ρ/c ≥ 1. M(a, b)
Then we have iK (T, Vρ ) = 0. Proof. Let e(t) ≡ 1 for t ∈ [0, 1]. Then e ∈ K. We prove that u 6= T (u) + λe for all u ∈ ∂Vρ
and λ > 0.
NONLOCAL IMPULSIVE BVPS
7
In fact, if this does not happen, there exist u ∈ ∂Vρ and λ > 0 such that u = T u + λe. We have, for all t ∈ [a, b] u(t) = γ(t)α[u] +
Z
1
k(t, s)g(s)f (s, u(s)) ds + γ(t)χ(τ,1]
0
I(u(τ )) +λ 1−τ
Z 1 I(u(τ )) + k(t, s)g(s)f (s, u(s)) ds + λ = γ(t) α[u] + 1−τ 0 Z 1 δ1 u(τ ) ≥ γ(t) α[u] + + k(t, s)g(s)f (s, u(s)) ds + λ 1−τ 0 Z 1 = γ(t)α1 [u] + k(t, s)g(s)f (s, u(s)) ds + λ
0
≥ c2 kγkα0 ρ + ρ
Z
b
k(t, s)g(s)fρ,ρ/c ds + λ
a
≥ ρ c2 kγkα0 +
1 fρ,ρ/c + λ. M(a, b)
By (I0ρ ), this implies that min u(t) ≥ ρ c2 kγkα0 +
t∈[a,b]
1 fρ,ρ/c + λ ≥ ρ + λ > ρ, M(a, b)
contradicting the fact that u ∈ ∂Vρ .
Note that, by means of the two Lemmas above, one may also provide a result on the existence of multiple positive solutions. In fact, if the nonlinearity f has a suitable oscillatory behavior, by nesting in an appropriate way several Vρ ’s and Kρ ’s, one may establish the existence of multiple positive solutions (we refer the reader to [16, 21] to see the type of results that may be stated). Here, for brevity, we state a result for the case of one positive solution for Eq. (2.1). Theorem 2.5. Eq. (2.1) has a positive solution in K if either of the following conditions hold. (H1 ) There exist ρ1 , ρ2 ∈ (0, ∞) with ρ1 < ρ2 such that (I1ρ1 ), (I0ρ2 ) hold. (H2 ) There exist ρ1 , ρ2 ∈ (0, ∞) with ρ1 < cρ2 such that (I0ρ1 ), (I1ρ2 ) hold. We omit the proof which follows simply from properties of fixed point index, for details of similar proofs see [15, 20]. Remark 2.6. So far we have discussed the case of having the impulse in just one point τ ∈ (0, 1). Similar arguments work in the case of a finite number of impulses. For example, in the case of two impulses, say ∆u|t=τ1 = I1 (u(τ1 )), ∆u′ |t=τ1 =
I1 (u(τ1 )) , τ1 − 1
∆u|t=τ2 = I2 (u(τ2 )), ∆u′ |t=τ2 =
I2 (u(τ2 )) , τ2 − 1
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GENNARO INFANTE AND PAOLAMARIA PIETRAMALA
where 0 < τ1 < τ2 < 1, one may work in the space (with an abuse of notation) P C[0, 1] := {u : [0, 1] → R : u is continuous in t ∈ [0, 1]\{τ1 , τ2 }, there exist u(τi− ) = u(τi ) and u(τi+ ) < ∞, i = 1, 2}, and seek for fixed points of the operator Z 1 I (u(τ )) I (u(τ )) 1 1 2 2 T˜u(t) := γ(t) α[u] + χ(τ1 ,1] + + χ(τ2 ,1] k(t, s)g(s)f (s, u(s)) ds. 1 − τ1 1 − τ2 0 in the cone (2.2), where [a, b] this time is a subset of (τ2 , 1). Thus, if there exist positive constants δ1,1 , δ1,2 , δ2,1 , δ2,2 such that δ1,i x ≤ Ii (x) ≤ δ2,i x for x ∈ [0, ∞) and i = 1, 2, ˜1 and dB ˜2 , where dB ˜1 is the Dirac measure of weight one may consider the measures dB ˜2 is the Dirac measure of weight δ1,1 /(1 − τ1 ) in τ1 and of weight δ1,2 /(1 − τ2 ), and dB δ2,1 /(1 − τ1 ) in τ1 and of weight δ2,2 /(1 − τ2 ) in τ2 . The above can be used to provide a modified version of Lemmas (2.3)-(2.4) in this new context. 3. Positive solutions of the impulsive BVP. We now consider the BVP u′′ (t) + g(t)f (t, u(t)) = 0, t ∈ (0, 1), t 6= τ,
(3.1)
∆u|t=τ = I(u(τ )), ∆u′ |t=τ =
(3.2) (3.3)
I(u(τ )) , τ −1
u(0) = α[u], u(1) = 0,
and we associate to this BVP the integral equation Z 1 I(u(τ )) , (3.4) u(t) = γ(t)α[u] + k(t, s)g(s)f (s, u(s)) ds + γ(t)χ(τ,1] 1−τ 0 where γ(t) = 1 − t for all t ∈ [0, 1], and k(t, s) =
s(1 − t),
t(1 − s),
s≤t s > t.
By a solution of the BVP (3.1)-(3.3) we mean a solution u ∈ P C[0, 1] of the corresponding integral equation (3.4). Here may choose Φ(s) = s(1 − s). Therefore, we can take [a, b] ⊂ (τ, 1) and (3.5)
c := min{a, 1 − b}.
NONLOCAL IMPULSIVE BVPS
9
Now (C2 ), (C3 ), (C4 ) are satisfied, (2.8) reads more simply (3.6)
c2 α0 + fρ,ρ/c ·
1 ≥ 1, M(a, b)
and (2.5) reads 1 Z 1 1 0,ρ A0 f ≤ 1. K(s)g(s) ds + + ρ(1 − Γ) 1−Γ 0 m
(3.7)
Example 3.1. We now assume that g ≡ 1, α[u] = αu(ξ), with α > 0 and ξ ∈ (τ, 1). In this case we have δ1 δ2 u(τ ), α2 [u] = αu(ξ) + u(τ ). 1−τ 1−τ We may take A0 = 0 and dA2 the Dirac measure of weight α in ξ and of weight δ2 /(1 − τ ) α1 [u] = αu(ξ) +
in τ . Thus Γ := and
Z
1
γ(t) dA2 (t) = α(1 − ξ) + δ2 0
1
δ2 α ξ(1 − ξ) + τ. 2 2 0 We can take [a, b] such that ξ ∈ [a, b]. Then we can set α0 = α, since α1 [u] ≥ αu(ξ) ≥ αρ Z
K(s) d(s) =
for u ∈ ∂Vρ . In this case (3.6) reads (3.8)
(1 − b)α + fρ,ρ/c ·
1 ≥ 1, M(a, b)
and (3.7) reads
(3.9)
1 0,ρ αξ(1 − ξ) + δ2 τ f ≤ 1. + 2 − 2(α(1 − ξ) + δ2 ) m
We now show that all the constants that appear in (3.8) and (3.9) can be computed. If we take τ = 1/5, ξ = 1/2, α = 4/5, δ2 = 1/2, we can choose [a, b] = [1/4, 3/4]. This gives m = 8, M(a, b) = 16 and c = 1/4. Therefore our requirements are fρ,ρ/c ≥ f 0,ρ ≤
64 5
and
8 . 13
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NONLOCAL IMPULSIVE BVPS
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[email protected] ` della Paolamaria Pietramala, Dipartimento di Matematica ed Informatica, Universita Calabria, 87036 Arcavacata di Rende, Cosenza, Italy E-mail address:
[email protected]
